If is a function, the * second
derivative* of y (or of f) is the derivative of the first
derivative. Notation:

Thus,

* Example.* Find the second derivatives of the
following functions.

(a) .

(b) .

(c) .

(a)

(b)

(c)

The first derivative gives information about whether a function increases or decreases. In fact:

(a) A differentiable function increases on intervals where its derivative is positive, and vice versa.

(b) A differentiable function decreases on intervals where its derivative is negative, and vice versa.

A function is * concave up* on an
open interval if is positive on the interval. And a function
is * concave down* on an open
interval if is negative on the interval.

A point where the concavity goes from up to down or from down to up
is called an * inflection point*.

What do these conditions mean geometrically?

Consider the curve below.

The tangent line A has negative slope, the tangent line B has zero
slope, and the tangent line C has positive slope. Therefore, as you
move from left to right, the slope of the tangent line
*increases*.

But the slope of the tangent line is given by , and to say something increases means its derivative is positive. So the derivative of --- which is --- must be positive. By the definition, this means the curve is concave up.

Now consider the curve below.

The tangent line at A has positive slope, the tangent line at B has zero slope, and the tangent line at C has negative slope. As you move from left to right, the slope of the tangent line decreases.

The slope of the tangent line is given by , and to say something decreases means its derivative is negative. So the derivative of --- which is --- must be negative. By the definition, this means the curve is concave down.

The two pictures exemplify the geometric meanings of * concave up* and * concave
down*.

* Example.* The graph of a function is pictured
below.

Determine the intervals on which the function is concave up and the intervals on which it is concave down. Find the x-coordinates of any inflection points.

The graph is concave up on , , and . The graph is concave down on .

Note that concavity is a property of a graph on an open interval, so the endpoints aren't included.

There are inflection points at and at .

* Example.* Find the intervals on which is concave up and the
intervals on which it is concave down. Find the x-coordinates of any
inflection points.

I set up a sign chart for , just as I use a sign chart for to tell where a function increases and where it decreases. The break points for my concavity sign chart will be the x-values where and the x-values where is undefined.

In this case, for and , and there are no points where is undefined. The break points are at and .

I picked numbers in each interval and plugged the numbers into . If is positive, I put a "+" on the interval and draw a concave-up curve below the interval; if is negative, I put a "-" on the interval and draw a concave-down curve below the interval.

The function is concave up for and for . It is concave down for . and are inflection points.

* Example.* Find the intervals on which is concave up and the intervals
on which it is concave down. Find the x-coordinates of any inflection
points.

for ; is undefined for .

The function is concave up for and for . It is concave down for . and are inflection points.

Concavity provides way to tell whether a critical point is a max or a
min --- well, sometimes. This method is called the *
Second Derivative Test*.

Consider a critical point where , i.e. where the tangent line is horizontal. Here are two possibilities.

The point A is a local max; it occurs at a place where the curve is concave down, i.e. where .

The point B is a local min; it occurs at a place where the curve is concave up, i.e. where .

* Theorem.* Suppose is defined on an open
interval, and for some point c in the interval . Then:

(a) If , then is a local max.

(b) If , then is a local min.

(c) If , the test fails. Try the First Derivative Test.

* Example.* Use the Second Derivative Test to
classify the critical points of .

The critical points are and .

Here's the graph:

In fact, is neither a max nor a min.

* Remark.* It is *not* true that if (so c is a critical point) and (so the Second Derivative Test fails), then
is neither a max nor a min. To say the test fails means that you can
draw *no conclusion*, and you need to do more work. The point
could *still* be a max or a min!

For example, consider . Then and , so and . Thus, is a critical point, and the Second Derivative Test fails. Nevertheless, is a local min, as you can verify by using the First Derivative Test.

This example also shows that if , it does not mean that c is an inflection point. In fact, the graph of is always concave up, so the concavity does not change at .

Copyright 2018 by Bruce Ikenaga